Boundary Control and Parameter Estimation

Overview¶

The previous lectures focused mostly on distributed controls, where the control acts as a volume source term. This lecture moves to two situations where the same optimal-control structure remains visible, but the functional analytic details become more delicate:

• controls acting on the boundary;

• unknown coefficients or material parameters.

The main new point is that the control-to-state map no longer has the same regularity as in the distributed case. Boundary controls interact with trace operators, while parameter controls enter the differential operator itself. Both effects change the correct space in which gradients and optimality conditions should be interpreted.

The logical path is:

• formulate Neumann boundary control in weak form;

• prove well-posedness by the trace theorem and Lax-Milgram;

• derive the adjoint equation and the reduced gradient using a Lagrangian;

• discuss Dirichlet boundary control through liftings;

• formulate coefficient identification as a PDE-constrained inverse problem;

• prove the sensitivity equation for the coefficient-to-state map;

• derive the adjoint gradient for parameter estimation from a Lagrangian;

• interpret box constraints as variational inequalities and projection formulas when enough regularity is available.

Throughout the lecture, let $\Omega\subset\mathbb R^d$ be a bounded Lipschitz domain, and let $\Gamma:=\partial\Omega$.

Boundary Controls and Trace Spaces¶

For a distributed control, the state equation typically has the form

where $B$ maps the control space into a volume dual space such as $H^{-1}(\Omega)$.

For a boundary control, the action of $u$ is instead mediated by the trace operator. The fundamental mapping is

which is continuous. Since $H^{1/2}(\Gamma)$ embeds continuously into $L^2(\Gamma)$ on bounded Lipschitz boundaries, there is a constant $C_{\mathrm{tr}}>0$ such that

This estimate is the key reason why Neumann data in $L^2(\Gamma)$ define a bounded linear functional on $H^1(\Omega)$:

Thus Neumann boundary control is naturally compatible with weak formulations. Dirichlet boundary control is subtler, because prescribing $y=u$ on $\Gamma$ requires $u$ to be a trace of an $H^1(\Omega)$ function, hence

is the natural energy space.

A Coercive Neumann Boundary Control Model¶

To avoid the compatibility condition of the pure Neumann Laplacian, we first consider the coercive elliptic equation

where $\sigma>0$. The pure Neumann case $\sigma=0$ is discussed below.

Let

and define

The weak state equation is:

find $y\in V$ such that

For a desired state $y_d\in L^2(\Omega)$ and a regularization parameter $\alpha>0$, the boundary control problem is

subject to the weak state equation.

The admissible set is assumed to be nonempty, closed, and convex in $L^2(\Gamma)$. A standard example is the box-constrained set

where $u_a,u_b\in L^\infty(\Gamma)$ and $u_a\le u_b$ a.e.

Well-Posedness of the Neumann State Equation¶

The bilinear form $a$ is continuous on $V\times V$ because

It is also coercive on $H^1(\Omega)$:

The right-hand side is bounded on $V$. Indeed,

and the trace estimate gives

By Lax-Milgram, for every $u\in L^2(\Gamma)$ there exists a unique state $y(u)\in H^1(\Omega)$ and

Moreover, the control-to-state map

is affine and continuous. If $f$ is fixed, its derivative is the linear map $S'(u)h=z$, where $z\in H^1(\Omega)$ solves

For the pure Neumann equation $\sigma=0$, constants belong to the kernel of the operator. One either works in the quotient space $H^1(\Omega)/\mathbb R$, or fixes the representative by imposing zero mean:

On $V_0$, Poincare’s inequality gives coercivity of $a(v,v)=\|\nabla v\|_{L^2(\Omega)}^2$. If the equation is required to hold for all test functions in $H^1(\Omega)$, the data must also satisfy the compatibility condition

This is why adding a positive reaction term is often the cleaner model for unconstrained Neumann boundary control.

Reduced Functional and Existence of an Optimal Control¶

Since the state is uniquely determined by $u$, define the reduced functional

The estimate above implies that $j$ is continuous on $L^2(\Gamma)$. The term $\frac\alpha2\|u\|_{L^2(\Gamma)}^2$ makes $j$ coercive:

Because $S$ is affine and continuous, the tracking term $\frac12\|S(u)-y_d\|_{L^2(\Omega)}^2$ is weakly lower semicontinuous. The regularization term is weakly lower semicontinuous as well. Therefore, by the direct method of the calculus of variations, $j$ attains a minimizer on every nonempty closed convex set $U_{\mathrm{ad}}\subset L^2(\Gamma)$.

Since $\alpha>0$, the reduced functional is strictly convex. Hence the optimal control is unique.

Adjoint Equation for Neumann Boundary Control¶

We derive the adjoint through the Lagrangian formalism. The weak state equation is the constraint

Introduce the Lagrangian

where $p\in V$ is the Lagrange multiplier. The sign in front of the constraint is a convention; this choice gives the gradient $\alpha u+\gamma p$.

Stationarity with respect to $p$ gives the state equation:

Stationarity with respect to $y$ gives the adjoint equation. For every state variation $\eta\in V$,

Thus

The adjoint problem is well posed by the same Lax-Milgram argument. If the solution is smooth, this weak equation corresponds to

Finally, stationarity with respect to the control gives the derivative in the control direction. For $h\in L^2(\Gamma)$,

Because the state and adjoint stationarity equations cancel all implicit state variations, this is the derivative of the reduced functional:

Thus the $L^2(\Gamma)$-gradient of the reduced functional is

If one puts the opposite sign in front of the constraint in the Lagrangian, the adjoint variable changes sign and the same optimality system can be written with $\alpha u-\gamma p$ instead.

Variational Inequality and Projection Formula¶

Let $\bar u\in U_{\mathrm{ad}}$ be the optimal control, with state $\bar y$ and adjoint $\bar p$. Since $U_{\mathrm{ad}}$ is closed and convex, the first-order necessary and sufficient condition is

Using the gradient formula, this becomes

For box constraints, the variational inequality is equivalent to the pointwise projection formula

Indeed, for almost every boundary point $x\in\Gamma$, the scalar condition is

This says exactly that

where $N_C$ denotes the normal cone to the convex set $C$. The normal-cone identity for projections gives the formula above.

Equivalently:

• if $u_a<\bar u<u_b$, then $\alpha\bar u+\gamma\bar p=0$;

• if $\bar u=u_a$, then $\alpha\bar u+\gamma\bar p\ge0$;

• if $\bar u=u_b$, then $\alpha\bar u+\gamma\bar p\le0$.

Dirichlet Boundary Control¶

For Dirichlet boundary control, the state equation is

Now the control cannot be an arbitrary element of $L^2(\Gamma)$ if the state is required to lie in $H^1(\Omega)$. The natural space is

Let $V_0:=H_0^1(\Omega)$. By the trace theorem, there exists a continuous lifting operator

such that

Write

The weak equation for $w$ is

Since $a$ is coercive on $H_0^1(\Omega)$, Lax-Milgram gives a unique $w\in H_0^1(\Omega)$, hence a unique state $y=w+Ru$.

The estimate is

Thus the Dirichlet control-to-state map is continuous from $H^{1/2}(\Gamma)$ into $H^1(\Omega)$.

Dirichlet Gradient and Regularity Warning¶

For Dirichlet control, it is useful to use the lifted variable $y=w+Ru$, with $w\in H_0^1(\Omega)$. The weak constraint is

The Lagrangian is

where $p\in H_0^1(\Omega)$ is the Lagrange multiplier. Stationarity with respect to $p$ gives the lifted state equation. Stationarity with respect to $w$ gives, for every $\eta\in H_0^1(\Omega)$,

Therefore the adjoint equation is

The control derivative is obtained by differentiating the same Lagrangian with respect to $u$. For $h\in U=H^{1/2}(\Gamma)$,

The subtle point is that $Rh$ is generally not an element of $H_0^1(\Omega)$, because its trace is $h$. Therefore one cannot simply insert $Rh$ as a test function in the adjoint equation, which is valid only for zero-trace variations.

Assume for a moment that $p$ is smooth enough for Green’s identity to be used classically. Since the adjoint satisfies

we compute

Integrating the first term by parts gives

Using $\gamma(Rh)=h$ and the strong adjoint equation, this becomes

Hence the two non-regularization terms in the Lagrangian control derivative reduce to the boundary term

Hence the control stationarity condition is naturally written as

where $R_U:U\to U'$ is the Riesz map of $H^{1/2}(\Gamma)$.

This is the key distinction from Neumann control:

• Neumann control with $u\in L^2(\Gamma)$ gives an $L^2(\Gamma)$ gradient involving the trace $\gamma p$;

• Dirichlet control gives a boundary gradient involving the normal derivative $\partial_n p$, which is generally only an element of $H^{-1/2}(\Gamma)$.

An $L^2(\Gamma)$ projection formula for Dirichlet control is therefore a formal or additional-regularity statement. It is valid, for instance, if $\partial_n p\in L^2(\Gamma)$ and the control is regularized in $L^2(\Gamma)$:

Without such regularity, the correct condition is the variational inequality in the duality pairing between $H^{-1/2}(\Gamma)$ and $H^{1/2}(\Gamma)$.

Parameter Estimation Problem¶

We now consider an inverse problem in which the unknown is a coefficient in the PDE. Let

For each admissible coefficient $q$, the state $y=y(q)\in H_0^1(\Omega)$ is defined by

In strong form this corresponds to

The coefficient identification problem is to choose $q$ so that $y(q)$ fits observed data $y_d$. A standard Tikhonov functional is

The admissible set is often taken as

with $0<q_a\le q_b$ a.e.

For every $q\in Q_{\mathrm{ad}}$, the bilinear form

is continuous and coercive on $H_0^1(\Omega)$:

Lax-Milgram therefore gives a unique state $y(q)$ and the stability estimate

Sensitivity Equation for the Coefficient¶

The coefficient-to-state map is nonlinear. We derive its derivative.

Let $\delta q\in L^\infty(\Omega)$ be a perturbation such that $q+t\delta q$ remains uniformly positive for small $t$. Let

Subtracting the two weak state equations gives

Divide by $t$ and pass formally to the limit $t\to0$. The sensitivity

solves

In strong form,

The equation is well posed because the right-hand side is bounded on $H_0^1(\Omega)$:

Thus

This proves that the derivative maps bounded coefficient perturbations to state perturbations continuously.

Adjoint Gradient for Parameter Estimation¶

We again derive the adjoint and the gradient through the Lagrangian. The weak state constraint is

Introduce

Stationarity with respect to $p$ gives the state equation. Stationarity with respect to $y$ gives, for every $\eta\in H_0^1(\Omega)$,

Therefore

Equivalently, in strong form,

The coefficient derivative is

Since the state and adjoint equations are precisely the stationarity conditions in $p$ and $y$, this is the derivative of the reduced functional:

If the product $\nabla y\cdot\nabla p$ belongs to $L^2(\Omega)$, then the $L^2(\Omega)$-gradient is

In the minimal energy setting, $\nabla y,\nabla p\in L^2(\Omega)^d$, so the product is a priori only in $L^1(\Omega)$. Therefore, the formula is always valid as a duality expression, while interpreting it as an $L^2$ gradient requires additional regularity or a discrete finite element setting.

Box Constraints for Parameter Estimation¶

Let $\bar q\in Q_{\mathrm{ad}}$ be a local minimizer, with state $\bar y$ and adjoint $\bar p$. The first-order condition is the variational inequality

Using the adjoint formula, this reads

When the gradient is an $L^2(\Omega)$ function, this is equivalent to the pointwise projection formula

The corresponding active-set interpretation is:

• on the inactive set $q_a<\bar q<q_b$,

  $$\alpha(\bar q-q_{\mathrm{ref}}) - \nabla\bar y\cdot\nabla\bar p=0;$$

  (84)

• on the lower active set $\bar q=q_a$,

  $$\alpha(\bar q-q_{\mathrm{ref}}) - \nabla\bar y\cdot\nabla\bar p\ge0;$$

  (85)

• on the upper active set $\bar q=q_b$,

  $$\alpha(\bar q-q_{\mathrm{ref}}) - \nabla\bar y\cdot\nabla\bar p\le0.$$

  (86)

Existence and Regularization in Inverse Problems¶

For distributed and Neumann controls, the regularization term $\frac\alpha2\|u\|_{L^2}^2$ gives coercivity in the same Hilbert space in which the control is optimized.

For coefficient estimation, the situation is more subtle:

• the PDE requires $q\in L^\infty(\Omega)$ with a positive lower bound;

• the Tikhonov term above controls only the $L^2(\Omega)$ distance from $q_{\mathrm{ref}}$;

• weak convergence of coefficients does not automatically imply strong convergence of the corresponding fluxes $q\nabla y$.

In a finite element discretization, existence is automatic because the admissible set is finite-dimensional and closed. In an infinite-dimensional analysis, one usually strengthens the compactness by choosing, for example,

or a bounded-variation regularization. These choices improve compactness and make the inverse problem stable with respect to noisy data.

This is the essential role of regularization: it is not merely a numerical penalty, but part of the mathematical formulation that turns an ill-posed inverse problem into a well-posed optimization problem.

Numerical Structure¶

Both classes of problems lead to the same computational loop:

• solve the state equation;

• solve the adjoint equation;

• compute the reduced gradient;

• apply a descent, Newton, or active-set update;

• enforce the admissible constraints.

For Neumann boundary control, the discrete gradient contains a boundary mass matrix because the control inner product is on $\Gamma$:

For Dirichlet boundary control, the discrete gradient involves the discrete normal derivative of the adjoint. This is more sensitive to mesh quality and boundary approximation.

For parameter estimation, the gradient at a quadrature point has the form

Thus the quality of the gradient depends on the accuracy of both state and adjoint gradients. This is why coefficient identification problems often require stabilization, mesh refinement, and careful treatment of noisy data.

Summary¶

Boundary control and parameter estimation fit into the same reduced optimization framework as distributed control, but the spaces change:

• Neumann control uses trace terms and gives gradients involving $\gamma p$;

• Dirichlet control uses liftings and gives dual boundary gradients involving $\partial_n p$;

• parameter estimation differentiates the PDE operator itself and gives gradients involving $\nabla y\cdot\nabla p$.

The main mathematical lessons are:

• the weak formulation determines the correct control space;

• Lax-Milgram gives the state and adjoint equations once coercivity and boundedness are checked;

• the reduced gradient is obtained by differentiating the Lagrangian with respect to the control or parameter after imposing state and adjoint stationarity;

• projection formulas are pointwise versions of variational inequalities;

• inverse problems require regularization not only for algorithms, but also for stability and well-posedness.

Suggested Reading¶

Tröltzsch¶

• Boundary control for elliptic PDEs

• Dirichlet and Neumann control

• Variational inequalities

Manzoni-Quarteroni-Salsa¶

• Boundary observation and control

• Parameter identification

• Inverse problems

De Los Reyes¶

• Reduced formulation

• Adjoint equations

• Regularization